Yes, both teams would send in a playbook to another person for the game to be ran. Most leagues have the games run by another coach or the league commish so that there is an impartial gamerunner. Really it is an honor system and there is nothing to stop someone from cheating besides the fact that if you are caught than you likely will be removed from the league and every other league likely would hear about it.

That third party coach is not a coach in the same division as either of the coches involvd in the game... plus once the playoffs start the game are run by coaches with nothing to gain in the game to keep things "clean"

Correct, no manual play calling. You fill in an AI punchcard for each play in your playbook. It's pretty easy, and the results are usually fun to watch. We have been running games live via join.me. Works great.